Chapter 5 - Hypothesis Tests with Means of Samples

Why did we use a comparison distribution of individual scores in previous chapters?

because we had sample sizes of one

What type of distribution should we use for sample sizes greater than one?

a distribution of means

distribution of means

a distribution of means of samples of a given size from a population

What are distributions of means theoretically based on?

a very large number of samples of the same size, which each sample randomly drawn from the same population of individuals

Rule 1 (distribution of means)

the mean of the distribution of means is equal to the mean of the population of individuals

Rule 2a (distribution of means)

the variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample

Rule 2b (distribution of means)

the standard deviation of a distribution of means is the square root of the variance of the distribution of means (standard error)

Rule 3 (distribution of means)

the shape of a distribution of means is approximately normal if either (a) each sample consists of 30 or more individuals or (b) the distribution of the population of individuals is normal

What are the 3 rules for distribution of means based on?

the Central Limit Theorem

population’s distribution

has scores of all individuals in the population

shape of population distribution

any; often normal

mean of population distribution

µ

variance of population distribution

σ²

standard deviation of population distribution

σ

sample distribution

has scores of the individuals in a single sample

shape of sample distribution

any

mean of sample distribution

M = (∑X)/N

variance of sample distribution

SD² = [∑(X-M)²]/N

standard deviation of sample distribution

SD = √SD²

distribution of means

means of samples randomly taken from the population

shape of distribution of means

approximately normal

mean of distribution of means

µ_M = µ

variance of distribution of means

σ²_M = σ² / N

standard deviation of distribution of means

σ_M = √σ²_M

What is the formula for the Z score for the sample’s mean on the distribution of means?

Z = (M - µ_M) / σ_M

Step 1 (hypothesis testing with a z test)

restate the question as research and null hypotheses about the populations

Step 2 (hypothesis testing with a z test)

determine the characteristics of the comparison distribution

Step 3 (hypothesis testing with a z test)

determine the cutoff sample score on the comparison distribution

Step 4 (hypothesis testing with a z test)

determine your sample’s score on the comparison distribution

Step 5 (hypothesis testing with a z test)

decide whether to reject the null hypothesis

marginal significance

when a result does not make the cutoff value at the usual 5% level but comes very close (say, p < 0.10)

When might marginal significance be appropriate to report?

when related results are clearly significant

What does reporting marginal results vary by?

the specialty areas in psychology

How frequently are z tests reported in research articles?

not often

How frequently are means and standard deviations reported in research articles?

typically

How might standard error be included in research articles?

as error bars

What is the best estimate of the population mean when it is unknown?

the sample mean

confidence interval (CI)

the range of scores that is likely to include the true population means

confidence limits

upper and lower ends of a confidence interval

What are typical confidence intervals?

95% and 99%

Step 1 (finding confidence limits)

figure the standard error

σ_M = √σ²_{M =} √σ² / N

Step 2 (finding confidence limits)

figure raw scores for the appropriate number of standard errors above/below the sample mean

• 95% → 1.96 standard errors

• 99% → 2.58 standard errors

upper limit

M + (SE)(σ_M)

lower limit

M - (SE)(σ_M)

How do we interpret a confidence interval?

we say we are __% confident that the true mean is in this range

What is confidence similar to?

subjective interpretation of probability

What can confidence intervals be used for?

hypothesis testing

What are the advantages of using confidence intervals rather than significance tests?

• give additional information

• focus attention on estimation

• less likely to be misused by researchers

Where are confidence intervals encouraged?

in the 2020 APA Manual

What are significance tests necessary for?

some advanced statistical procedures

How are confidence intervals reported?

following mean, in brackets

ex: 95% CI [62.06, 73.20]

How many standard errors are 95% confidence intervals in both directions?

approximately 2 (±1.96)

How many standard errors are 99% confidence intervals in both directions?

approximately 2.5 (±2.58)

What do error bars in research articles typically represent?

standard errors, but about 15% of the time, they’re actually confidence intervals